While Salinger certainly showed a preference for a white cover, it’s highly unlikely he has any actual say in the cover design.
The first paperback edition has a picture of Holden talking to a woman who was clearly a prostitute – she was wearing revealing clothing and leaning against a lamppost. Salinger was furious and told his publisher to give it a white cover. However, the publisher was under no obligation to do so (authors very rarely get any contractual say in the design of the book covers, and since the book has remained in print, the original contract is still in force*). However, in response to both Salinger’s complaint and the realization that the book wasn’t just going to be a throwaway paperback, Bantam switched to the plain red cover.
Eventually, the book was given a plain white cover. This coincided with the release of Salinger’s other books in uniform editions, and was probably influenced by his antipathy to the original paperback cover. It’s highly unlikely it was because of the assassins (and the fact that is was in their possession is the Frederick Wertham fallacy – millions of others had the book in their possession at the time, and never killed anyone).
It’s not true Salinger won’t allow any books to be published with cover art; current hardcovers and trade paperbacks of Catcher reproduce the original cover art of a red horse.
*Publishing contracts are typically written to be in force for a set period (usually seven years). If, at the end of that period, the book is not in print, the author can ask the publisher to revert the rights. Since Catcher never went out of print, the original contract should still apply.